ScalingStacks

Consider $j\in\{1,\ldots,n-1\}$. Fix $k\in\{0,\ldots,r\}$ such that $i_{k}\leq j<i_{k+1}$. Let us show that

(3.2.2)

$$\gamma_{I}t_{j}=\begin{cases}t_{j+r-k}\gamma_{I}&\text{ if }i_{k}<j<i_{k+1}-1\\ 0&\text{ if }i_{k}=j<i_{k+1}-1\\ \gamma_{\{i_{1}<\cdots<i_{k}<i_{k+1}-1<i_{k+2}<\cdots<i_{r}\}}&\text{ if }i_{k}<j=i_{k+1}-1\\ t_{k}\gamma_{I}&\text{ if }i_{k}=j=i_{k+1}-1.\end{cases}$$